Solutions for the Constant Quantum Yang-Baxter Equation From Lie (Super) Algebras

Abstract

We present a systematic procedure to obtain singular solutions of the constant quantum Yang-Baxter equation in arbitrary dimension. This approach, inspired by the Lie (super)algebra structure, is explicitly applied to the particular case of (graded) contractions of the orthogonal real algebra ${\mathfrak{so}}(N+1)$. In this way we show that “classical” contraction parameters which appear in the commutation relations of the contracted Lie algebras, become quantum deformation parameters, arising as entries of the resulting quantum $R$-matrices.